HITTING TIME STATISTICS AND EXTREME VALUE THEORY

Transcription

1 HITTING TIME STATISTICS AND EXTREME VALUE THEORY ANA CRISTINA MOREIRA FREITAS, JORGE MILHAZES FREITAS, AND MIKE TODD Abstract. We consider discrete time dynamical systems and show the link between Hitting Time Statistics the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa. We apply these results to non-uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws with no extra condition on the speed of mixing, for example). We also give applications of our theory to higher dimensional examples, for which we also obtain classical extreme value laws and exponential hitting time statistics for balls). We extend these ideas to the subsequent returns to asymptotically small sets, linking the Poisson statistics of both processes. 1. Introduction In this paper we demonstrate and exploit the link between Extreme Value Laws EVL) and the laws for the Hitting Time Statistics HTS) for discrete time non-uniformly hyperbolic dynamical systems with an absolutely continuous invariant measure. The setting is a discrete time dynamical system X, B, µ, f), where X is a d-dimensional Riemannian manifold, B is the Borel σ-algebra, f : X X is a measurable map and µ an f-invariant probability measure for all A B we have µf 1 A)) = µa)). We consider a Riemannian metric on X that we denote by dist and for any ζ X and δ > 0, we define B δ ζ) = {x X : distx, ζ) < δ}. Also let Leb denote Lebesgue measure on X and for every A B we will write A := LebA). The measure µ will be an absolutely continuous invariant probability measure acip) with density denoted by ρ = dµ. We will denote dleb R + := 0, ) and R + 0 := [0, ) Extreme Value Laws. In this context, by EVL we mean the study of the asymptotic distribution of the partial maximum of observable random variables evaluated along the orbits of the system. To be more precise, take an observable ϕ : X R {± } achieving Date: April 6, Mathematics Subject Classification. 37A50, 37C40, 60G10, 60G70, 37B20, 37D25, 37E05. Key words and phrases. Return Time Statistics, Extreme Value Theory, Non-uniform hyperbolicity, Interval maps. JMF is partially supported by POCI/MAT/61237/2004 and MT is supported by FCT grant SFRH/BPD/26521/2006. All three authors are supported by FCT through CMUP. 1

2 2 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD a global maximum at ζ X we allow ϕζ) = + ) and consider the stationary stochastic process X 0, X 1,... given by Define the partial maximum X n = ϕ f n, for each n N. 1.1) M n := max{x 0,..., X n 1 }. 1.2) If µ is ergodic then Birkhoff s law of large numbers says that M n ϕζ) almost surely. Similarly to central limit laws for partial sums, we are interested in knowing if there are normalising sequences {a n } n N R + and {b n } n N R such that µ {x : a n M n b n ) y}) = µ {x : M n u n }) Hy), 1.3) for some non-degenerate distribution function d.f.) H, as n. Here u n := u n y) = y a n + b n 1.4) is such that nµx 0 > u n ) τ, as n, 1.5) for some τ = τy) 0 and in fact Hy) = Hτy)). For more details of the existence of such sequences u n, see Lemma 2.1.) When this happens we say that we have an Extreme Value Law EVL) for M n. Note that, clearly, we must have u n ϕζ), as n. We refer to an event {X j > u n } as an exceedance, at time j, of level u n. Classical Extreme Value Theory asserts that there are only three types of non-degenerate asymptotic distributions for the maximum of an independent and identically distributed i.i.d.) sample under linear normalisation. They will be referred to as classical EVLs and we denote them by: Type 1: EV 1 y) = e e y for y R; this is also known as the Gumbel extreme value distribution e.v.d.). Type 2: EV 2 y) = e y α, for y > 0, EV 2 y) = 0, otherwise, where α > 0 is a parameter; this family of d.f.s is known as the Fréchet e.v.d. Type 3: EV 3 y) = e y)α, for y 0, EV 3 y) = 1, otherwise, where α > 0 is a parameter; this family of d.f.s is known as the Weibull e.v.d. The same limit laws apply to stationary stochastic processes, under certain conditions on the dependence structure, which allow the reduction to the independent case. With this in mind, to a given stochastic process X 0, X 1,... we associate an i.i.d. sequence Y 0, Y 1,..., whose d.f. is the same as that of X 0, and whose partial maximum we define as ˆM n := max{y 0,..., Y n 1 }. 1.6) We want to compare the asymptotic distribution of M n with that of ˆMn, when properly normalised. We recall that because Y 0, Y 1,... are i.i.d., by [LLR, Theorem 1.5.1], the convergence in 1.5) is equivalent to µ ˆM n u n ) = µx 0 u n )) n e τ, as n. 1.7)

3 HITTING TIMES AND EXTREME VALUES 3 Depending on the type of limit law that applies, we have that τ = τy) is of one of the following three types: τ 1 y) = e y for y R, τ 2 y) = y α for y > 0, and τ 3 y) = y) α for y 0. In the dependent context, the general strategy is to prove that if X 0, X 1,... satisfies some conditions, then the same limit law for ˆMn applies to M n with the same normalising sequences {a n } n N and {b n } n N. Following [LLR] we refer to these conditions as Du n ) and D u n ), where u n is the sequence of thresholds appearing in 1.3). Both conditions impose some sort of independence but while Du n ) acts on the long range, D u n ) is a short range requirement. The original condition Du n ) from [LLR], which we will denote by D 1 u n ), is a type of uniform mixing requirement specially adapted to Extreme Value Theory. Let F i1,...,i n x 1,..., x n ) denote the joint d.f. of X i1,..., X in, and set F i1,...,i n u) = F i1,...,i n u,..., u). Condition D 1 u n )). We say that D 1 u n ) holds for the sequence X 0, X 1,... if for any integers i 1 <... < i p and j 1 <... < j k for which j 1 i p > m, and any large n N, Fi1,...,i p,j 1,...,j k u n ) F i1,...,i p u n )F j1,...,j k u n ) γn, m), where γn, m n ) 0, for some sequence m n = on). Since usually the information concerning mixing rates of the systems is known through decay of correlations, in [FF2] we proposed a weaker version, which we will denote by D 2 u n ), which still allows us to relate the distributions of ˆMn and M n. The advantage is that it follows immediately from sufficiently fast decay of correlations for observables which are of bounded variation or Hölder continuous see [FF2, Section 2] and Lemma 6.1). Condition D 2 u n )). We say that D 2 u n ) holds for the sequence X 0, X 1,... if for any integers l, t and n µ {X 0 > u n } {max{x t,..., X t+l 1 } u n }) µ{x 0 > u n })µ{m l u n }) γn, t), where γn, t) is nonincreasing in t for each n and nγn, t n ) 0 as n for some sequence t n = on). By 1.5), the sequence u n is such that the average number of exceedances in the time interval {0,..., n/k } is approximately τ/k, which goes to zero as k. However, the exceedances may have a tendency to be concentrated in the time period following the first exceedance at time 0. To avoid this we introduce: Condition D u n )). We say that D u n ) holds for the sequence X 0, X 1,... if n/k lim lim sup n µ{x 0 > u n } {X j > u n }) = ) k j=1

4 4 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD This guarantees that the exceedances should appear scattered through the time period {0,..., n 1}. The main result in [FF2, Theorem 1] states that if D 2 u n ) and D u n ) hold for the process X 0, X 1,... and for a sequence of levels satisfying 1.5), then the following limits exist, and lim µ ˆM n u n ) = lim µm n u n ). 1.9) The above statement remains true if we replace D 2 u n ) by D 1 u n ) see [LLR, Theorem 3.5.2]). We assume that the observable ϕ : X R {+ } is of the form ϕx) = gdistx, ζ)), 1.10) where ζ is a chosen point in the phase space X and the function g : [0, + ) R {+ } is such that 0 is a global maximum g0) may be + ); g is a strictly decreasing bijection g : V W in a neighbourhood V of 0; and has one of the following three types of behaviour: Type 1: there exists some strictly positive function p : W R such that for all y R g1 1 s + yps)) lim s g 1 0) g1 1 = e y ; 1.11) s) Type 2: g 2 0) = + and there exists β > 0 such that for all y > 0 g2 1 sy) lim s + g2 1 s) = y β ; 1.12) Type 3: g 3 0) = D < + and there exists γ > 0 such that for all y > 0 g3 1 D sy) lim s 0 g3 1 D s) = yγ. 1.13) Examples of each one of the three types are as follows: g 1 x) = log x in this case 1.11) is easily verified with p 1), g 2 x) = x 1/α for some α > 0 condition 1.12) is verified with β = α) and g 3 x) = D x 1/α for some D R and α > 0 condition 1.13) is verified with γ = α). Remark 1. Let the d.f. F be given by F u) = µx 0 u) and set u F = sup{y : F y) < 1}. Observe that if at time j N we have an exceedance of the level u sufficiently large), i.e., X j x) > u, then we have an entrance of the orbit of x into the ball B g 1 u)ζ) of radius g 1 u) around ζ, at time j. This means that the behaviour of the tail of F, i.e., the behaviour of 1 F u) as u u F is determined by g 1, if we assume that Lebesgue s Differentiation Theorem holds for ζ, since in that case 1 F u) ρζ) B g 1 u)ζ), where ρζ) = dµ ζ). From classical Extreme Value Theory we know that the behaviour of the dleb tail determines the limit law for partial maximums of i.i.d. sequences and vice-versa. The above conditions are just the translation in terms of the shape of g 1, of the sufficient and necessary conditions on the tail of F of [LLR, Theorem 1.6.2], in order to exist a

5 HITTING TIMES AND EXTREME VALUES 5 non-degenerate limit distribution for ˆMn. In fact, if some EV i applies to ˆM n, for some i {1, 2, 3}, then g must be of type g i. As can be seen from the definitions of D 2 u n ) and D u n ), proving EVLs for absolutely continuous invariant measures for uniformly expanding dynamical systems is straightforward. The study of EVLs for non-uniformly hyperbolic dynamical systems has been addressed in the papers [Col2] and [FF1]. In [Col2], Collet considered non-uniformly hyperbolic C 2 maps of the interval which admit an acip µ, with exponential decay of correlations and obtained a Gumbel EVL for observables of type g 1 actually he took g 1 x) = log x), achieving a global maximum at µ-a.e. ζ in the phase space. We remark that neither the critical points nor its orbits were included in this full µ-measure set of points ζ. In [FF1] the quadratic maps f a x) = 1 ax 2 on I = [ 1, 1] were considered, with a BC, where BC is the Benedicks-Carleson parameter set introduced in [BC]. For each map f a with a BC, a Weibull EVL was obtained for observables of type g 3 achieving a maximum either at the critical point or at the critical value Hitting Time Statistics. We next turn to Hitting Time Statistics for the dynamical system X, B, f, µ). For a set A X we let r A y) denote the first hitting time to A of the point y. That is, the first time j 1 so that f j y) A. We will be interested in the fluctuations of this functions as the set A shrinks. Firstly we consider the Return Time Statistics RTS) of this system. Let µ A denote the conditional measure on A, i.e., µ A := µ A. By Kac s Lemma, the expected value of r µa) A with respect to µ is r A A dµ A = 1/µA). So in studying the fluctuations of r A on A, the relevant normalising factor is 1/µA). Given a sequence of sets {U n } n N so that µu n ) 0, the system has Return Time Statistics Gt) for {U n } n N if for all t 0 the following limit exists and equals Gt): lim µ U n r Un t µu n ) ). 1.14) We say that X, f, µ) has Return Time Statistics Gt) to balls at ζ if for any sequence {δ n } n N R + such that δ n 0 as n we have RTS Gt) for U n = B δn ζ). If we study r A defined on the whole of X, i.e., not simply restricted to A, we are studying the Hitting Time Statistics. Note that we will use the same normalising factor 1/µA) in this case. Analogously to the above, given a sequence of sets {U n } n N so that µu n ) 0, the system has Hitting Time Statistics Gt) for {U n } n N if for all t 0 the following limit is defined and equals Gt): lim µ r Un t ). 1.15) µu n ) HTS to balls at a point ζ is defined analogously to RTS to balls. In [HLV], it was shown that the limit for the HTS defined in 1.15) exists if and only if the limit for the analogous

6 6 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD RTS defined in 1.14) exists. Moreover, they show that the HTS distribution exists and is exponential i.e., Gt) = e t ) if and only if the RTS distribution exists and is exponential. For many mixing systems it is known that the HTS are exponential around almost every point. For example, this was shown for Axiom A diffeomorphisms in [H], transitive Markov chains in [Pi] and uniformly expanding maps of the interval in [Col1]. Note that in these papers the authors were also interested in the Poisson) statistics of subsequent returns to some shrinking sets. For various results on some systems with some strong hyperbolicity properties see also e.g. [Ab, C1, AG]. Note that for systems with good mixing properties, but at some special periodic) points, [H] obtained similar distributions for the HTS/RTS with an exponential parameter 0 < θ < 1 i.e., Gt) = e θt ); while for the sequence of successive returns to neighbourhoods of these points a compound Poisson distribution was proved in [HV]. For non-uniformly hyperbolic systems less is known. A major breakthrough in the study of HTS/RTS for non-uniformly hyperbolic maps was made in [HSV], where they gave a set of conditions which, when satisfied, imply exponential RTS to cylinders and/or balls. Their principal application was to maps of the interval with an indifferent fixed point. They also provided similar conditions to imply Poisson) laws for the subsequent visits of points to shrinking sets. See Section 5). Another important paper in this direction was [BSTV], in which they showed that the RTS for a map are the same as the RTS for the first return map. The first return map to a set U X is the map F = f r U.) Since it is often the case that the first return maps for non-uniformly hyperbolic dynamical systems are much better behaved possibly hyperbolic) than the original system, this provided an extremely useful tool in this theory. For example, they proved that if f : I I is a unimodal map for which the critical point is nowhere dense, and for which an acip µ exists, then the relevant first return systems U, F, µ U ) have a Rychlik property. They then showed that such systems, studied in [R], must have exponential RTS, and hence the original system I, f, µ) also has exponential RTS to balls around µ-a.e. point). The presence of a recurrent critical point means that the first return map itself will not satisfy this Rychlik property. To overcome this problem in [BV] special induced maps, U, F ), were used, where for x U we have F x) = f indx) x) for some inducing time indx) N that is not necessarily the first return time of x to U. The fact that these particular maps can be seen as first return maps in the canonical Markov extension, the Hofbauer tower, meant that they were still able to exploit the main result of [BSTV] to get exponential RTS around µ-a.e. point for unimodal maps f : I I with an acip µ as long as f satisfies a polynomial growth condition along the critical orbit. In [BT] this result was improved to include any multimodal map with an acip, irrespective of the growth along the critical orbits, and of the speed of mixing. We would like to remark that in the case of partially hyperbolic dynamical systems, [Do] proved exponential RTS, using techniques similar to [Pi]. In fact the theory there also

7 HITTING TIMES AND EXTREME VALUES 7 covers the Poisson) statistics of subsequent returns to shrinking sets of balls. These statistics were also considered for toral automorphisms, using a different method, in [DGS]. We note that for dynamical systems X, B, f, µ) where µ is an equilibrium state, the RTS/HTS to the dynamically defined cylinders are often well understood, see for example [AG]. However, for non-uniformly hyperbolic dynamical systems it is not always possible to go from these strong results to the corresponding results for balls. We would like to emphasise that in this paper we focus on the HTS to balls, rather than cylinders Main Results. Our first main result, which obtains EVLs from HTS, is the following. Theorem 1. Let X, B, µ, f) be a dynamical system where µ is an acip, and consider ζ X for which Lebesgue s Differentiation Theorem holds. If we have HTS to balls centred on ζ X, then we have an EVL for M n which applies to the observables 1.10) achieving a maximum at ζ. If we have exponential HTS Gt) = e t ) to balls at ζ X, then we have an EVL for M n which coincides with that of ˆMn meaning that 1.9) holds). In particular, this EVL must be one of the 3 classical types. Moreover, if g is of type g i, for some i {1, 2, 3}, then we have an EVL for M n of type EV i. We next define a class of multimodal interval maps f : I I. We denote the finite set of critical points by Crit. We say that c Crit is non-flat if there exists a diffeomorphism ψ c : R R with ψ c 0) = 0 and 1 < l c < such that for x close to c, fx) = fc)± ψ c x c) lc. The value of l c is known as the critical order of c. Let NF k := { f : I I : f is C k, each c Crit is non-flat and } inf Df n p) > 1. f n p)=p The following is a simple corollary of Theorem 1 and [BT, Theorem 3]. It generalises the result of Collet in [Col2] from unimodal maps with exponential growth on the critical point to multimodal maps where we only need to know that there is an acip. Corollary 1. Suppose that f NF 2 and f has an acip µ. Then I, f, µ) has an EVL for M n which coincides with that of ˆMn, and this holds for µ-a.e. ζ X fixed at the choice of the observable in 1.10). Moreover, the EVL is of type EV i when the observables are of type g i, for each i {1, 2, 3}. Now, we state a result in the other direction, i.e., we show how to get HTS from EVLs. Theorem 2. Let X, B, µ, f) be a dynamical system where µ is an acip and consider ζ X for which Lebesgue s Differentiation Theorem holds. If we have an EVL for M n which applies to the observables 1.10) achieving a maximum at ζ X then we have HTS to balls at ζ. If we have an EVL for M n which coincides with that of ˆMn, then we have exponential HTS Gt) = e t ) to balls at ζ.

8 8 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD The following is immediate by the above and [FF2, Theorem 1] see 1.9)). Corollary 2. Let X, B, µ, f) be a dynamical system where µ is an acip and consider ζ X for which Lebesgue s Differentiation Theorem holds. If D 2 u n ) or D 1 u n )) and D u n ) hold for a stochastic process X 0, X 1,... defined by 1.1) and 1.10), where u n is a sequence of levels satisfying 1.5), then we have exponential HTS to balls at ζ. The following is an immediate corollary of Theorem 2 and the main theorem of [FF1]. Corollary 3. For every Benedicks-Carleson quadratic map f a with a BC) we have exponential HTS to balls around the critical point or the critical value. The next result is a byproduct of Theorems 1, 2 and the fact that under D 1 u n ) the only possible limit laws for partial maximums are the classical EV i for i {1, 2, 3}. Since this is not as immediate as the other corollaries, we include a short proof in Section 2. Corollary 4. Let X, B, µ, f) be a dynamical system µ is an acip and consider ζ X for which Lebesgue s Differentiation Theorem holds. If D 1 u n ) holds for a stochastic process X 0, X 1,... defined by 1.1) and 1.10), where u n is a sequence of levels satisfying 1.5), then the only possible HTS to balls around ζ are of exponential type, meaning that, there is θ > 0 such that Gt) = e θt. Note that for this corollary to be non-trivial, we must assume that there exists a distribution for HTS. This may not always be the case. For example, in [CF, C2] it was shown that for certain circle diffeomorphisms there are sequences of intervals {U n } n N, {V n } n N which both shrink to the same point ζ, but yield different HTS laws. Note that in these cases D 1 u n ) also fails. Remark 2. In this paper we are concerned with observables of the form ϕx) = gdistx, ζ)) where ζ is some typical point for our measure µ and g is a particular kind of function which is adapted to give different types of EVL. In our proof of the link between HTS and EVL for such observables it is useful to be able to express µb ɛ ζ)) in terms of ɛ d where d is the dimension of the space. For measures µ other than acips, this is highly problematic. log µb In many cases, for a measure µ, the limit lim ɛ ζ)) ɛ 0 exists for µ-a.e. ζ in fact this log ɛ constant can be referred to as the dimension of the measure [P]). However, removing the logs in this expression, we would typically expect the liminf and limsup to be 0 and infinity respectively even for well behaved systems. This occurs for example for the doubling map f : x 2x mod 1 when µ is the α, 1 α)-bernoulli measure and α 1/2. This follows from the fact that the measure of sets can be computed via certain Birkhoff averages, using the Gibbs property, and the fact that these averages should typically fluctuate, which can be seen from the Law of the Iterated Logarithm in this case, see [DP]. We therefore expect serious intrinsic problems in proving results for general measures for this kind of observable with measures which are not acips. On the other hand, if, given a system X, f, µ), we are prepared to change our observable to one which reflects properties of µ, i.e. replacing gdistx, ζ)) with gψx, ζ)) for some ψ, then it is possible to prove analogous results to those presented here. This is the subject of forthcoming work.

9 HITTING TIMES AND EXTREME VALUES 9 As we have already mentioned, Corollary 1 generalises the result of Collet in [Col2], which was for C 2 non-uniformly hyperbolic maps of the interval admitting a Young tower). However, a close look to Collet s arguments allows us to conclude that his result still prevails in higher dimensions. In fact, one can show that if we consider non-uniformly expanding maps in any finite dimensional compact manifold), admitting a so-called Young tower with exponential return times to the base, then for any sequence of r.v. X 0, X 1,..., defined as in 1.1) and for a sequence of levels u n such that nµx 0 > u n ) τ > 0, conditions D 2 u n ) and D u n ) hold. This means that by the above theorems, we can prove both EVLs and HTS for these maps. Due to numerous definitions required for that setting, we leave both the theorems and the proofs on this subject to Section 6. Theorems 1 and 2 give us new tools to investigate the recurrence of dynamical systems, principally by allowing us to use the wealth of theory for HTS which has been developed in recent years to prove EVLs. We note that in Corollary 1, the dynamical systems involved need not have any fast rate of decay of correlations at all. Indeed, a priori the relevant system may only have summable decay of correlations. As in Section 6 where we consider higher dimensional maps admitting Young towers, there are situations where it is actually easier to check conditions like D 2 u n ) and D u n ) in order to get laws for HTS. In fact, to our knowledge, exponential HTS to balls have never been proved before for higher dimensional non-uniformly expanding systems: in such cases, inducing schemes with the nice properties of one-dimensional dynamics are much harder to find. Also the dynamical systems we present in this paper should provide models which can be used in investigating Extreme Value Theory both analytically and numerically. Namely, the simple fact that we get EVLs from deterministic models may be an extra advantage for numerical simulation since there is no need to generate random numbers. This means that this theory may reveal very useful for testing GEV Generalised Extreme Value distribution) fitting for data corresponding to phenomena for which there is an underlying deterministic model. The next question that arises is: what about subsequent visits to U n or subsequent exceedances of the level u n? Namely, we are interested in the point processes associated to the instants of occurrence of returns to U n and exceedances of the level u n. If we have either exponential HTS or a classical EVL then time between hits or exceedances is exponentially distributed. This means that we should expect a Poisson limit for the point processes. We show in Section 3 that the relation between HTS and EVL does indeed extend to the laws for the subsequent visits/exceedances we postpone the precise definitions and results to Sections 3, 4 and 5). More precisely, we show that the point process of hitting times has a Poisson limit if and only if the point process of exceedances has a Poisson limit. We next discuss how to obtain a Poisson law in these two different contexts. In Section 4 we give conditions which guarantee a Poisson limit for the point process of exceedance times. This part of the paper can be seen as a generalisation of [FF2]. Moreover, we show that these conditions can be verified in the settings from [Col2, FF1], leading to Poisson statistics for both point processes for the systems considered. In Section 5 we show that in many cases for multimodal maps it can be shown that the HTS behave asymptotically as a Poisson distribution.

10 10 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD In a parallel work [HNT], EVLs have also been proved for dynamical systems with acips. For example they proved EVLs for flows, and also considered observables ϕ with multiple maxima. We point out that in the setting of multimodal maps and observables ϕ of the form ϕx) = gdx, ζ)) where ζ is some typical point of an acip µ, our Corollary 1 improves on their results since we do not require any knowledge of the decay of correlations. Throughout this paper the notation A n B n means that lim A n B n = 1. Also, if {δ n } n N R + has δ n 0 as n, then for each ζ X, let κ 0, ) be such that B δn ζ) κ δ d n. Let x R. We denote the integer part of x by x and define x := x if x = x, and x := x + 1 otherwise. Acknowledgements. We would like to thank J.F. Alves for useful suggestions regarding the example of a non-uniformly expanding system given in Section 6.2. We thank the referee for useful comments. 2. Proofs of our results on HTS and EVL In this section we prove Theorems 1, 2 and Corollary 4. Proof of Theorem 1. Let ρζ) = dµ ζ) dleb R+ 0 and set u n = g ) 1 κρζ)n) 1/d + p g 1 κρζ)n) 1/d)) y d, for y R, for type g 1; u n = g 2 κρζ)n) 1/d) y, for y > 0, for type g 2 ; u n = D D g 3 κρζ)n) 1/d)) y), for y < 0, for type g 3. We remark that the choices of the normalising sequences {a n } n N and {b n } n N, which determine the sequence of levels {u n } n N as in 1.4), are made accordingly to the behaviour of the tail of d.f. F, given by F u) = µx 0 u). [LLR, Corollary 1.6.3] gives a canonical choice in terms of the tail of F. In fact, up to linear scaling the normalising sequences are unique by Khintchine s Theorem see [LLR, Theorem 1.2.3]). Since, as we have explained in Remark 1, the shape of g 1 determines the tail of the d.f. F, the definitions of the levels u n above are just a reflection of this fact. For n sufficiently large we have n 1 n 1 {x : M n x) u n } = {x : X j x) u n } = {x : gdistf j x), ζ)) u n } j=0 n 1 = {x : distf j x), ζ) g 1 u n )} = {x : r Bg 1 u n) ζ) x) n} 2.1) j=0 j=0

11 HITTING TIMES AND EXTREME VALUES 11 Now, observe that 1.11), 1.12) and 1.13) imply g1 1 u n ) = g1 [g 1 1 κρζ)n) 1/d) + p g 1 κρζ)n) 1/d)) y ] d g1 [g 1 1 κρζ)n) 1/d)] ) e e y/d y 1/d = ; κρζ)n g2 1 u n ) = g2 [g 1 2 κρζ)n) 1/d) ] y g2 [g 1 2 κρζ)n) 1/d)] ) y y β βd 1/d = ; κρζ)n [ g3 1 u n ) = g3 1 D D g 3 κρζ)n) 1/d)) ] y) Thus, we may write meaning that [ g3 1 D D g 3 κρζ)n) 1/d))] y) γ = g 1 u n ) ) 1/d τy), κρζ)n ) 1/d g 1 τi y) i u n ), i {1, 2, 3} κρζ)n ) y) γd 1/d. κρζ)n where τ 1 y) = e y for y R, τ 2 y) = y βd for y > 0, and τ 3 y) = y) γd for y < 0. Since Lebesgue s Differentiation Theorem holds for ζ X, we have µb δζ)) B δ ζ) ρζ) as δ 0. Consequently, since it is obvious that g 1 u n ) 0 as n, then µ B g 1 u n)ζ) ) ρζ) B g 1 u n)ζ) ρζ)κg 1 u n )) d = ρζ)κ τy) κρζ)n = τy) n. Thus, we have n τy) µ ). 2.2) B g 1 u n)ζ) Now, we claim that using 2.1) and 2.2), we have { lim µ{x : M nx) u n }) = lim µ which gives the first part of the theorem. x : r Bg 1 un) ζ)x) }) τy) µ B g 1 u n )ζ) ) 2.3) = Gτy)), 2.4)

12 12 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD To see that 2.3) holds, observe that by 2.1) and 2.2) we have { }) µ{m τy) n u n }) µ r Bg 1 u n) ζ) µ B ) g 1 u n)ζ) { }) { = µ r Bg 1 un) ζ) n µ r Bg 1 un) ζ) 1 + ε n )n}), where {ε n } n N is such that ε n 0 as n. Since we have { } { } m+k 1 r Bg 1 u n) ζ) m \ r Bg 1 u n) ζ) m + k f j B g 1 u n )ζ) ), m, k N, 2.5) j=m it follows by stationarity that { }) { µ r Bg 1 un) ζ) n µ r Bg 1 un) ζ) 1 + ε n )n}) as n, completing the proof of 2.3). ε n nµ B g 1 u n )ζ) ) ε n τ 0, Next we will use the exponential HTS hypothesis, that is Gt) = e t, to show the second part of the theorem. Under the exponential HTS assumption, by 2.4) it follows immediately that lim µ{x : M n x) u n }) = e τy). Recall that in the corresponding i.i.d setting, i.e. when we are considering {x : ˆMn x) u n } rather than {x : M n x) u n }, 1.5) is equivalent to 1.7). Therefore we also have lim µ{x : ˆMn x) u n }) = e τy), since nµx 0 > u n ) = nµ B g 1 u n )ζ) ) τy), as n. As explained in the introduction, this means that in the i.i.d. setting Gτ) must be of the three classical types. It remains to show that if the observable is of type g i then lim µ{x : M n x) u n }) = e τy) means that the EVL that applies to M n rather than ˆM n ) is also of type EV i, for each i {1, 2, 3}. Type g 1 : In this case we have e τ 1y) = e e y, for all y R, that corresponds to the Gumbel e.v.d. and so we have an EVL for M n of type EV 1. Type g 2 : We obtain e τ2y) = e y βd for y > 0. To conclude that in this case we have the Fréchet e.v.d. with parameter βd, we only have to check that for y 0, µ{x : M n x) u n }) = 0. Since g 2 κρζ)n) 1/d) > 0 for all large n) and µ{x : M n x) u n }) = µ {x : M n x) g 2 κρζ)n) 1/d) }) y e y βd as n. Letting y 0, it follows that µ{x : M n x) 0}) 0, and, for y < 0, µ{x : M n x) u n }) = µ {x : M n x) g 2 κρζ)n) 1/d) }) y µ{x : M n x) 0}) 0. So, we have, in this case, an EVL for M n of type EV 2.

13 HITTING TIMES AND EXTREME VALUES 13 Type g 3 : For y < 0, we have e τ 3y) = e y)γd. To conclude that in this case we have the Weibull e.v.d. with parameter γd, we only need to check that for y 0, µ{x : M n x) u n }) = 1. In fact, for y 0, since D g 3 κρζ)n) 1/d) > 0, we have { µ{x : M n x) u n }) = µ x : M n x) D g 3 κρζ)n) 1/d)) }) y + D µ{x : M n x) D}) = 1. So we have, in this case, an EVL for M n of type EV 3. For the proof of Theorem 2, we will require the following lemma. This is essentially contained in [LLR, Theorem 1.6.2], but we give a proof for completeness. Lemma 2.1. Suppose that X, B, f, µ) is a dynamical system where µ is an acip. Furthermore, let ϕx) = gdistx, ζ)) for some ζ X where g is one of the three types described above. Then, for each y R, there exists a sequence {u n y)} n N as in 1.4) such that nµ{x : ϕx) > u n y)}) τy) 0. Moreover, for every t > 0 there exists y R such that τy) = t. Proof. We will prove the lemma in the case when g is of type g 2. For the other two types of g, the argument is the same, but with minor adjustments, see [LLR, Theorem 1.6.2]. First we show that we can always find a sequence {γ n } n N such that nµx 0 > γ n ) 1. Take γ n := inf{y : µx 0 y) 1 1/n}, and let us show that it has the desired property. Note that nµx 0 > γ n ) 1, which means that lim sup nµx 0 > γ n ) 1. Using 1.12), for any z < 1, we have lim inf µx 0 > γ n ) µx 0 > γ n z) = lim inf ρζ)κ g 1 2 γ n ) ) d ρζ)κ g 1 2 zγ n ) ) d = lim inf g 1 2 γ n ) ) d z β g 1 2 γ n ) ) d = z βd. Since, by definition of γ n, for any z < 1, nµx 0 > γ n z) 1, letting z 1, it follows immediately that lim inf nµx 0 > γ n ) 1. Now let u n y) = γ n y, which means that, for all n N, we are taking a n = γ 1 n and b n = 0 in 1.4). Then, using 1.12), it follows that for all y > 0 nµx 0 > γ n y) = nµ {x : distx, ζ) < g 1 2 γ n y)} ) nρζ)κ g 1 2 γ n y) ) d nρζ)κy β g 1 2 γ n )) d y βd nρζ)κ g 1 2 γ n ) ) d y βd nµx 0 > γ n ) y βd. So taking y = t 1/βd) > 0 would suit our purposes.

14 14 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD Proof of Theorem 2. We assume that by hypothesis for every y R and some sequence u n = u n y) as in 1.4) such that nµ {x : ϕx) > u n y)}) τy), we have lim µ {x : M nx) u n y)}) = Hτy)). Observe that, by Khintchine s Theorem see [LLR, Theorem 1.2.3]), up to linear scaling the normalising sequences are unique, which means that we may assume that they are the ones given by Lemma 2.1. Hence given t > 0, Lemma 2.1 implies that there exists y R such that nµ {x : ϕx) > u n y)}) t. Given {δ n } n N R + with δ n We will prove If n is sufficiently large, then 0, we define l n := t/κρζ)δ d n). g 1 u ln ) δ n. 2.6) {x : ϕx) > u n } = {x : gdistx, ζ)) > u n } = {x : distx, ζ) < g 1 u n )} = B g 1 u n)ζ). Hence, by assumption on the sequence u n, we have nµ B g 1 u n )ζ) ) τy) = t. As Lebesgue s Differentiation Theorem holds for ζ X, we have µb δζ)) B δ ρζ) as δ 0. ζ) Consequently, since it is obvious that g 1 u n ) 0 as n, then n B g 1 u n)ζ) 1/d t/ρζ). Thus, we may write g 1 t u n ) κnρζ)) and substituting n by ln we are immediately led to 2.6) by definition of l n. Next, using Lebesgue s Differentiation Theorem, again, we get µ B δn ζ)) ρζ)κδ d n which easily implies that by definition of l n, t µ B δn ζ)) l n. 2.7) Now we note that, as in 2.1) l n 1 l n 1 {x : M ln x) u ln } = {x : X j x) u ln } = {x : gdistf j x), ζ)) u ln } = l n 1 j=0 j=0 {x : distf j x), ζ) g 1 u ln )} = {x : r Bg 1 ul x) l n )ζ) n }. 2.8) j=0 At this point, we claim that lim µ { x : r Bδn ζ)x) }) t = lim µ {x : M ln x) u ln }). 2.9) µb δn ζ))

15 HITTING TIMES AND EXTREME VALUES 15 Then, the first part of the theorem follows, once we observe that, by hypothesis, we have µ {x : M ln x) u ln }) Hτy)) = Ht). For the second part of the theorem, first notice that for the i.i.d. setting, i.e. when we are considering {x : ˆMn x) u n } rather ) than {x : M n x) u n }, 1.5) is equivalent to 1.7). Therefore, µ {x : ˆMn x) u n } e τy) as n. Hence if the EVL of M n coincides with that of ˆMn, then we also have Hτy)) = e τy). It remains to show that 2.9) holds. First, observe that { }) t µ r Bδn ζ) = µ {M ln u ln }) + µ { }) r Bδn ζ) l n µ {Mln u ln }) ) µb δn ζ)) + µ { r Bδn ζ) For the third term on the right, note that by 2.7) we have µ { }) t r Bδn ζ) l n µ {r Bδn ζ) µb δn ζ))}) }) t µ { }) ) r Bδn ζ) l n. µb δn ζ)) = µ { rbδ n ζ) l n }) µ { rbδ n ζ) 1 + ε n )l n }), for some sequence {ε n } n N such that ε n 0, as n. By 2.5), 2.7) and stationarity it follows that µ { r Bδn ζ) l n }) µ { rbδn ζ) 1 + ε n )l n }) ε n l n µ B δn ζ)) ε n t 0, as n. For the remaining term, using 2.6), 2.7) and 2.8), we have µ { rbδ n ζ) l n }) µ {Mln u ln }) = µ { rbδ n ζ) l n }) µ {r Bg 1 uln ) ζ) l n }) as n, which ends the proof of 2.9). l n i=1 µ f i B δn ζ) B g 1 u l n ) ζ) )) = l n µ B δn ζ) B g 1 u l n ) ζ) ) t µ B δn ζ)) µ B g µ B δn ζ)) 1 u l n ) ζ) ) = t 1 µ Bg 1 u ln ) ζ) ) µ B δn ζ)) 0 Proof of Corollary 4. Let us assume the existence of HTS to balls around ζ not necessarily exponential). Then the first part of Theorem 1 assures the existence of an EVL as in 1.3)

16 16 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD for M n defined in 1.2). This fact and the hypothesis that D 1 u n ) holds allows us to use [LLR, Theorem 3.7.1] to conclude that there is θ > 0 such that lim µm n u n ) = e θτ. Finally, we use the first part of Theorem 2 to conclude that we have HTS to balls centred on ζ of exponential type. 3. Relation between hitting times and exceedance point processes We have already seen how to relate HTS and EVL. We next show that if we enrich the process and the statistics by considering either multiple returns or multiple exceedances we can take the parallelism even further. Given a sequence {δ n } n N R + such that δ n waiting or inter-hitting) time as and the j-th hitting time as w j B δn ζ) x) = r B δ n ζ) f w1 B δn ζ) x)+ +wj 1 B δn ζ) x) x) r j B δ n ζ)x) = j wb i δn ζ)x). i=1 0, for each j N, we define the j-th ), 3.1) We define the Hitting Times Point Process HTPP) by counting the number of hitting times during the time interval [0, t). However, since µb δn ζ)) 0, as n, then by Kac s Theorem, the expected waiting time between hits is diverging to as n increases. This fact suggests a time re-scaling using the factor v n := 1/µB δn ζ)), which is precisely the expected inter-hitting time. Hence, for any x X and every t 0 define { } v Nnt) = Nn[0, t), x) := sup j : r j B δn ζ) x) nt v nt = 1 Bδn ζ) f j 3.2) When x B δn ζ) and we consider the conditional measure µ Bδ n ζ) instead of µ, then we refer to N nt) as the Return Times Point Process RTPP). If we have exponential HTS, Gt) = e t in 1.14)), then the distribution of the waiting time before hitting B δn ζ) is asymptotically exponential. Also, if we assume that our systems are mixing, because in that case we can think that the process gets renewed when we come back to B δn ζ), then one may look at the hitting times as the sum of almost independent r.v.s that are almost exponentially distributed. Hence, one would expect that the hitting times, when properly re-scaled, should form a point process with a Poisson type behaviour at the limit. As discussed in Section 1.2, for hyperbolic systems, it is indeed the case that we do get a Poisson Process as the limit of HTPP. The theory in [HSV, BSTV] and [BT] implies that if f NF 2 has an acip then we have a Poisson limit for the HTPP. We postpone a sketch of this fact to Section 5, in order to keep our focus on the relation between HTS and EVL j=0

17 HITTING TIMES AND EXTREME VALUES 17 here. However, we would like to remark that a key difference between proofs for the first hitting time and for showing that we have a Poisson Point Process, if we are using the theory started in [HSV], is that a further mixing condition is required. Now, we turn to an EVL point of view. In this context, one is concerned with the occurrence of exceedances of the level u n for the stationary stochastic process X 0, X 1,.... In particular, we are interested in counting the number of exceedances, among a random sample X 0,..., X n 1 of size n. As in the previous sections, we consider the stationary stochastic process defined by 1.1) and a sequence of levels {u n } n N such that nµx 0 > u n ) τ > 0, as n. We define the exceedance point process EPP) by counting the number of exceedances during the time interval [0, t). We re-scale time using the factor v n := 1/µX > u n ) given by Kac s Theorem, again. Then for any x X and every t 0, set N n t) = N n [0, t), x) := v n t j=0 1 Xj >u n. 3.3) The limit laws for these point processes can be used to assess the impact and damage caused by rare events since they describe their time occurrences, their individual impacts and accumulated effects. Assuming that the process is mixing, we almost have a situation of many Bernoulli trials where the expected number of successes is almost constant nµx > u n ) τ > 0). Thus, we expect a Poisson law as a limit. In fact, one should expect that the exceedance instants, when properly normalised, should form a point process with a Poisson Process as a limit, also. This is the content of [LLR, Theorem 5.2.1] which states that under D 1 u n ) and D u n ), the EPP N n, when properly normalised, converges in distribution to a Poisson Process. See [LLR, Chapter 5], [HHL] and references therein for more information on the subject). Similarly to Theorems 1 and 2, we show that if there exists a limiting continuous time stochastic process for the HTPP, when properly normalised, then the same holds for the d EPP and vice-versa. In the sequel denotes convergence in distribution. Theorem 3. Let X, B, µ, f) be a dynamical system where µ is an acip and consider ζ X for which Lebesgue s Differentiation Theorem holds. Suppose that for any sequence δ n 0 we have that the HTPP defined in 3.2) is such that N d n N, where N is a continuous time stochastic process. Then, for the EPP defined in 3.3) we also have d N n N. Proof. The result follows immediately once we set δ n = g 1 u n ) and observe that for every j, n N and x X we have {x : X j > u n } = {x : f j x) B g 1 u n )ζ)}, which implies that N n t) = Nnt), for all t 0. Corollary 5. Suppose that f NF 2 and f has an acip µ. Then, denoting by N n the associated EPP as in 3.3), we have N d n N, as n, where N denotes a Poisson Process with intensity 1.

18 18 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD The fact that the maps in this corollary satisfy the conditions of Theorem 3 follows from the sketch in Section 5. So the result is otherwise immediate. Theorem 4. Let X, B, µ, f) be a dynamical system where µ is an acip and consider ζ X for which Lebesgue s Differentiation Theorem holds. Suppose that for a sequence of levels {u n } n N such that nµx 0 > u n ) τ > 0, as n, the EPP defined in 3.3) is such that N n d N, where N is a continuous time stochastic process. Then, for the HTPP defined in 3.3) we also have N n d N. Proof. Given a sequence {δ n } n N R + with δ n 0 we define, as in the proof of Theorem 2, the sequence l n such that δ n g 1 u ln ). Set k n := max{v n, v ln } and observe that N nt) N ln t) k n j=0 1 B δ n ζ) B g 1 u l n )ζ) f j. Using stationarity we get µ N nt) N ln t) > 0) k n µ B δn ζ) B g 1 u l n ) ζ) ) = k n µ B δn ζ)) µ B g 1 u l n ) ζ) ) 0, by definition of l n. The result now follows immediately by Slutsky s Theorem see [DM, Theorem ]). 4. Poisson Statistics via EVL As we have already mentioned, [LLR, Theorem 5.2.1] states that for a stationary stochastic process satisfying D 1 u n ) and D u n ), the EPP N n defined in 3.3) converges in distribution to a Poisson Process. The main result in [FF2] states that in order to prove an EVL for stationary stochastic processes arising from a dynamical system, it suffices to show conditions D 2 u n ) and D u n ). This proved to be an advantage over [LLR, Theorem 3.5.2] since the mixing information of systems is usually known through decay of correlations that can be easily used to prove D 2 u n ), as opposed to condition D 1 u n ) appearing in [LLR, Theorem 3.5.2]. Our goal here is to prove that we still get the Poisson limit if we relax D 1 u n ) so that it suffices to have sufficiently fast decay of correlations of the dynamical systems that generate the stochastic processes. However, for that purpose, one needs to strengthen D 2 u n ) in order to cope with multiple events. Something similar was necessary in the corresponding theory in [HSV].) For that reason we introduce condition D 3 u n ) below, that still follows from sufficiently fast decay of correlations, as D 2 u n ) did, and together with D u n ) allows us to obtain the Poisson limit for the EPP. Let S denote the semi-ring of subsets of R + 0 whose elements are intervals of the type [a, b), for a, b R + 0. Let R denote the ring generated by S. Recall that for every A R there are k N and k intervals I 1,..., I k S such that A = k i=1i j. In order to fix notation, let a j, b j R + 0 be such that I j = [a j, b j ) S. For I = [a, b) S and α R, we denote

19 HITTING TIMES AND EXTREME VALUES 19 αi := [αa, αb) and I +α := [a+α, b+α). Similarly, for A R define αa := αi 1 αi k and A + α := I 1 + α) I k + α). For every A R we define MA) := max{x i : i A Z}. In the particular case where A = [0, n) we simply write, as before, M n = M[0, n). At this point, we propose: Condition D 3 u n )). Let A R and t N. We say that D 3 u n ) holds for the sequence X 0, X 1,... if µ {X 0 > u n } {MA + t) u n }) µ{x 0 > u n })µ{ma) u n }) γn, t), where γn, t) is nonincreasing in t for each n and nγn, t n ) 0 as n for some sequence t n = on), which means that t n /n 0 as n. Recalling the definition of the EPP N n t) = N n [0, t) given in 3.3), we set N n [a, b) := Nb) Na) = v n b j= v n a 1 {Xj >u n }. We now state the main result of this section that gives the Poisson statistics for the EPP under D 3 u n ) and D u n ). Theorem 5. Let X 1, X 2,... be a stationary stochastic process for which conditions D 3 u n ) and D u n ) hold for a sequence of levels u n such that nµx 0 > u n ) τ > 0, as n. Then the EPP N n defined in 3.3) is such that N d n N, as n, where N denotes a Poisson Process with intensity 1. As a consequence of this theorem, Theorem 4 and the results in [FF1] we get: Corollary 6. For any Benedicks-Carleson quadratic map f a with a BC), consider a stochastic process X 0, X 1,... defined by 1.1) and 1.10), with ζ being either the critical point or the critical value. Then, denoting by N n the associated EPP as in 3.3), we have N d n N, as n, where N denotes a Poisson Process with intensity 1. Moreover, if we consider Nn, the HTPP as in 3.2), for balls around either the critical point or the critical value, then the same limit also applies to Nn. With minor adjustments to [Col2], we can use Theorem 5 to show that, similarly to Corollary 5, interval maps with exponential decay of correlations have Poisson statistics for the EPP. However, we will not state this result here, since we prove a more general result which works in higher dimensions) in Section 6.

20 20 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD 4.1. Proofs of the results. In this section we prove Theorem 5 and Corollary 6. The key is Proposition 1 whose proof we prepare with the following two Lemmas. These are very similar to ones in [FF2, Section 3] and [Col2, Section 3], so we omit the details of the proofs. However, note that in contrast to the lemmas in the papers mentioned above, we need them to take care of events that depend on nonconsecutive random variables. Lemma 4.1. For any l N and u R we have l 1 l 1 µx j > u) µm l > u) µx j > u) j=0 j=0 l 1 l 1 j=0 i=0,i j µ{x j > u} {X i > u}) Proof. This is a straightforward consequence of the formula for the probability of a multiple union of events. See for example the first Theorem of Chapter 4 in [Fe]. Lemma 4.2. Assume that r, s, l, t are nonnegative integers. Suppose that A, B R are such that A B. Set l := #{j N : j B \ A}. Assume that min{x : x A} r + t and let A 0 = [0, r + t). Then, we have 0 µma) u) µmb) u) l µx > u) 4.1) and µma r 1 0 A) u) µma) u) + µ {X > u} {MA i) u}) i=0 r 1 2r µ{x > u} {X i > u}) + tµx > u). 4.2) i=1 The proof of this lemma can easily be done by following the proof of [FF2, Lemma 3.2] with minor adjustments. Proposition 1. Let A R be such that that A = p j=1 I j where I j = [a j, b j ) S, j = 1,..., p and a 1 < b 1 < a 2 < < b p 1 < a p < b p. Let {u n } n N be such that nµx 0 > u n ) τ > 0, as n, for some τ 0. Assume that conditions D 3 u n ) and D u n ) hold. Then, µ M na) u n ) n + p µmni j ) u n ) = j=1 p e τb j a j ). Proof. Let h := inf j {1,...,p} {b j a j } and H := sup{x : x A}. Take k > 2/h and n sufficiently large. Note this guarantees that if we partition n[0, H] Z into blocks of length r n := n/k, J 1 = [Hn r n, Hn), J 2 = [Hn 2r n, Hn r n ),..., J Hk = [Hn Hkr n, n Hk 1)r n ), J Hk+1 = [0, Hn Hkr n ), then there is more than one of these blocks contained in ni i. Let S l = S l k) be the number of blocks J j contained in ni l, that is, S l := #{j {1,..., Hk} : J j ni l }. j=1

21 HITTING TIMES AND EXTREME VALUES 21 As we have already observed S l > 1 l {1,..., p}. For each l {1,..., p}, we define l A l := I p i+1. i=1 Set i l := min{j {1,..., k} : J j ni l }. Then J il, J il +1,..., J il +s l for each i {i p l+1,..., i p l+1 + S p l+1 } let ni l. Now, fix l and B i := i j=i p l+1 J j, J i := [Hn ir n, Hn i 1)r n t n ) and J i := J i J i. Note that J i = r n t n and J i = t n. See Figure 1 for more of an idea of the notation here. na l 0 Hn ni p-1 ni 1 ni 2 ni p ni p-l+1 J* i J i ' Ji p-l+1 +S p-l+1 J i Ji p-l+1 B i Figure 1. Notation

22 22 A. C. M. FREITAS, J. M. FREITAS, AND M. TODD We have, µmb i na l 1 ) u n ) 1 r n µx > u n ))µmb i 1 na l 1 ) u n ) = µmb i na l 1 ) u n ) µmb i 1 na l 1 ) u n ) + r n µx > u n )µmb i 1 na l 1 ) u n ) µmb i na l 1 ) u n ) µmb i 1 na l 1 ) u n ) + + r n t n 1 j=0 µ{x j+hn irn > u n } {MB i na l 1 ) u n )} r n t n )µx > u n )µmb i 1 na l 1 ) u n ) r n t n 1 j=0 + t n µx > u n ). µ{x j+hn irn > u n } {MB i na l 1 ) u n )} By Lemma 4.2, we obtain µmb i na l 1 ) u n ) 1 r n µx > u n ))µ MB i 1 na l 1 ) u n ) r n t n 1 2r n t n ) µ{x > u n } {X j > u n }) + t n µx > u n ) + r n t n 1 j=0 j=1 + t n µx > u n ), µx > u n )µmb i 1 na l 1 ) u n ) µ{x > u n } {MB i 1 na l 1 ) d j ) u n }) where d j = j + Hn ir n ). Now using condition D 3 u n ), we obtain µmb i na l 1 ) u n ) 1 r n µx > u n ))µmb i 1 na l 1 ) u n ) Set r n t n 1 2r n t n ) µ{x > u n } {X j > u n }) + 2t n µx > u n ) + r n t n )γn, t n ). j=1 r n t n 1 Υ k,n := 2r n t n ) µ{x > u n } {X j > u n }) + 2t n µx > u n ) + r n t n )γn, t n ). j=1

23 HITTING TIMES AND EXTREME VALUES 23 Recalling 1.5), we may assume that n and k are sufficiently large so that n k µx > u n) < 2 and 1 r n µx > u n ) < 1 which implies µmbsp l+1 na l 1 ) u n ) 1 r n µx > u n ))µmb Sp l+1 1 na l 1 ) u n ) Υk,n, and µmbsp l+1 na l 1 ) u n ) 1 r n µx > u n )) 2 µmb Sp l+1 2 na l 1 ) u n ) µmb Sp l+1 na l 1 ) u n ) 1 r n µx > u n ))µmb Sp l+1 1 na l 1 ) u n ) + 1 r n µx > u n ) µmb Sp l+1 1 na l 1 ) u n ) 1 r n µx > u n ))µmb Sp l+1 2 na l 1 ) 2Υ k,n. Inductively, we obtain µmbsp l+1 na l 1 ) u n ) 1 r n µx > u n )) S p l+1 µmna l 1 ) u n ) Sp l+1 Υ k,n. Using Lemma 4.2, µmnal ) u n ) 1 r n µx > u n )) S p l+1 µmna l 1 ) u n ) µmnal ) u n ) µmb Sp l+1 na l 1 ) u n ) + µmbsp l+1 na l 1 ) u n ) 1 r n µx > u n )) S p l+1 µmna l 1 ) u n ) µmni p l+1 na l 1 ) u n ) µm S p l+1 i=i l J i na l 1 ) u n ) + S p l+1 Υ k,n 2r n µx > u n ) + S p l+1 Υ k,n. In the next step we have µmna l ) u n ) 1 r n µx > u n )) S p l+1+s p l+2 µmna l 2 ) u n ) µmna l ) u n ) 1 r n µx > u n )) S p l+1 µmna l 1 ) u n ) + 1 r n µx > u n ) S p l+1 µmna l 1 ) u n ) 1 r n µx > u n )) S p l+2 µmna l 2 ) u n ) 4r n µx > u n ) + S p l+1 + S p l+2 )Υ k,n. Therefore, by induction, we obtain µmna p ) u n ) 1 r n µx > u n )) p j=1 S j 2pr n µx > u n ) + p S j Υ k,n. j=1